Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER IV. MATHEMATICS AND THE MODERN VIEW OF THE WORLD

28. AND THEN CAME EINSTEIN

Let us recall where matters stood toward the end of the nineteenth century with regard to the mathematical description of nature. On the one hand, Newton's mechanics based on arithmetic and Euclidean geometry had attained enormous success in both celestial mechanics and earthly engineering problems. The success of part of Newton's mechanics may be attributed to the existence of the mysterious matter, the ether. On the other hand, Maxwell had presented equations that predicted the existence of electromagnetic waves, which had indeed been found. The ether could not constitute the medium in which those waves moved, and no other medium was known. In addition, if Maxwell's equations were applied in Newtonian geometry, they lacked one very important element of Newton's theory—that the measurements could be taken in every inertial system. At the same time, mathematicians and physicists of the time started questioning whether Euclidean geometry was the right one for describing the world we live in. The feeling was, and Gauss stated it explicitly, that arithmetic could be relied upon as a tool for describing nature, but as it was not clear what the geometry of nature was, geometry could not be relied upon as such a tool.

This was the situation with which Einstein was familiar. Several additional discoveries and hypotheses were made that could have influenced him, but it is not clear to what extent he was exposed to and aware of them. One such discovery was the famous experiment performed by Albert Michelson and Edward Morley, two American physicists. At that time the scientific community still believed in the existence of the ether as the medium through which forces are exerted (Maxwell's theory had not as yet been accepted; the Michelson-Morley experiment took place before electromagnetic radiation was discovered in a laboratory). One of the questions that arose was: In what direction does the ether move relative to the Earth? The idea of the experiments was to utilize the fact that the ether was the medium through which light waves propagate.

The principle was simple. Assume that a beam of light travels from a source, A, on Earth toward a mirror at point B and back, with the mirror arranged such that the direction from A to B is the direction of the Earth's orbit. At the same time, another beam of light was sent from the same source A to a mirror at point C and back, with the direction from A to C at right angles to the direction of the Earth's orbit. A simple calculation shows that the first beam of light would get back to A after the second beam. A calculation is needed because the claim is not at all intuitive. In order to understand that it is correct, imagine that the speed of light is just twice the speed at which the Earth is moving. In that case, the distance that the first beam has to travel to get from A to B is twice the actual distance between them. In the time that takes, the second beam will have already gotten back to A; that is, it will get back to the source sooner. In reality, the speed of light is much faster than twice the speed of the Earth, and the time difference between each of the beam's returns to the source would be minimal.

To carry out the measurements required a series of sophisticated experiments, which Michelson started in 1881; Morley joined him in 1886, and their joint efforts showed that there was no difference between the times the beam arrived back at the source. This finding raised doubts regarding the hypothesis that the ether was the medium through which light propagated.

The Dutch physicist Hendrik Lorentz proposed a mathematical formula describing the dynamics that fitted the results of the Michelson-Morley experiment. The formula corrected the assumption that until then had been considered self-evident, as it was based on what our senses perceive. The assumption was that if an object F is moving at a velocity v₁ relative to object G, which is moving at a velocity v₂ relative to object H, then F is moving at a velocity of v₁ + v₂ relative to object H. Lorentz proposed replacing this formula with another (whose detailed formulation is not important for the current discussion) that showed that, at small speeds, the speed of F relative to H is very close to what our senses perceive, that is, v₁ + v₂, but when F is moving at very high speed relative to G, say close to the speed of light, and G is moving at a speed close to the speed of light relative to H, then the speed of Frelative to H is also close to the speed of light (and not twice that speed, as Newton's theory would infer). Lorentz also observed that if his formula is used in Maxwell's equations, the same equations are obtained in all inertial systems.

The French mathematician Henri Poincaré (1854–1912), who was then one of the most famous mathematicians in the world, stated that Lorentz's formulae gave the best explanation for the difference between Newton's laws and Maxwell's equations. He himself developed the mechanics deriving from Lorentz's transformation, and already in 1900 and later in several additional papers, he published a complete version of what later became known as the special theory of relativity, including a version of the connection between mass and energy, that is, E = mc². These developments of formulae presented by Lorentz were extremely close to Einstein's special theory of relativity, which we will describe below. Yet neither Lorentz, though a famous physicist who actually was awarded the Nobel Prize in Physics in 1902, nor Poincaré, nor others who tried with the help of those formulae to reconcile Newton's and Maxwell's theories drew the right conclusions from the formulae. It was Albert Einstein who did so.

Einstein (the extent of whose knowledge of these developments is unknown, as we stated above) adopted Lorentz's formulae and distilled from them a physical property that completely contradicts intuition based on our daily experiences. The property is that the speed of light is constant in every inertial system of coordinates. That is to say that if light travels at a speed of c relative to an object G, and G is traveling much faster than an object H, even at half the speed of light, the light will still travel at speed c relative to object H. This leads to the conclusion that it is not possible to get an object to move faster than the speed of light. In a certain sense, Einstein claimed that the geometry of the world, including formulae for combining speeds, is described better by Lorentz's formulae than by Newton's. As stated above, at relatively low speeds Lorentz's formulae are very close to Newton's, which is why our intuition, which is based on our daily experiences of only relatively low speeds, led us to accept Newton's formulae as describing the world.

The mathematics that describes the new relations between position and speed brought Einstein to the realization of the possibility of something never previously envisaged, and in particular something that was opposed to our intuition (in the wake of Maxwell's basic contribution, the realization that mathematics can lead to revelations of totally new phenomena had become accepted). The equation leads to the conclusion that mass is converted to energy. This was a mathematical statement that could have remained in the realm of a technical mathematical conclusion without any physical importance. Einstein, however, interpreted the equation as a physical truth and drew the conclusion that the substitution between mass and energy was possible (although at that point in time he did not see a way to control and exploit this possibility). He even derived the famous formula E = mc² (after several more-complex versions) from the equation for the mass-energy equivalence. Einstein published these discoveries in two papers that appeared in 1905. Three years later, in 1908, Hermann Minkowski (1864–1909), who was one of Einstein's tutors when the latter was studying at the Institute of Technology in Zurich, Switzerland, presented a new geometry in which the coordinate of time did not have special status but was to be treated like the other, spatial, coordinates. To formulate the rules of this new geometry, Minkowski developed what is called tensorial calculus, which extended Newton's system of derivatives to more complex relations. Thus the special theory of relativity became established from both the physical and the basic mathematical aspects.

An interesting question is why Einstein received all the credit for the special theory of relativity when, as noted above, its essence, including the formula E = mc², had been published by Poincaré before Einstein. If we ignore the conspiracy speculations that are raised from time to time, the answer has two parts. The first difference between Einstein's theory and Poincaré's is a conceptual one. Poincaré developed mathematics and did not notice, or at least did not declare and emphasize, that mathematics presents physics with new principles. Einstein focused on the physical principles from which the new mechanics are derived, of course with the help of mathematics. Therefore, to attribute the new physics to Einstein is completely justified. (Einstein worked on his theory while in the Bern patent office, before he learned of all of Poincaré's findings.) The second part of the answer is that Poincaré's papers are densely written and hard to read, whereas Einstein immediately concentrated on the essence and the new, and he presented his theory in an almost intuitive way. For example, Poincaré gave the mass/energy ratio as m = , an expression that is harder to grasp than the more familiar version. Simplicity has a clear advantage.

The special theory of relativity unified Newton's mechanics with the electromagnetic mechanics of Maxwell; in other words, it gave a joint mathematical system to both physical phenomena. The mathematical analysis of speeds significantly below the speed of light in effect coincides with Newton's theory. The effects related to the theory of relativity emerge only at speeds close to the speed of light. In classic engineering situations, the use of Newton's formulae is accurate enough, and for many years the part of the theory that related to relativity was in the domain only of scientists. In the current era, when for example communication at the speed of light is relevant to everyone, the equations that describe relativity are in widespread engineering use.

One Newtonian law of nature, gravity, remains outside the mathematical framework of special relativity. Moreover, Newton's second law and the law of gravity both relate to mass, the same mass. If these are different laws there is no reason for the same physical quantity to serve both. Einstein suggested that the two laws were two aspects of the same effect. The example he gave to make the point was a free fall in an elevator, that is, an elevator whose cable has broken and is falling freely, inside which, although a passenger is accelerating due to the force of gravity, he is not aware of any force being exerted. That is, the exertion of the force is also relative. Unlike the case of special relativity, where the mathematics had already been developed and Einstein's main contribution was to present the physical interpretation, in the analysis of gravity Einstein first posed the physical hypothesis. Without mathematics, however, the hypothesis had no scientific value. Einstein devoted several years of research to the attempt to find a mathematical theory that would unify gravity and the other forces, published a number of articles with partial results, and finally, in 1916, published the definitive paper presenting the general theory of relativity. Once again the solution required a new presentation of the geometry of the world. The mathematical framework Einstein used was that proposed by Riemann some sixty years earlier (see the previous section), and the mathematical tool that describes the mechanics in this geometric world was the tensor calculus that Minkowski developed to show the geometry of the world of special relativity.

Einstein adopted Galileo and Newton's idea of inertia, but he adopted the version that stated that a body that has no force exerted on it will continue to move along a geodesic, that is, the shortest line between two points in a space. Then Einstein claimed that in the geometry of the physical space, the shortest line between two points in this space is not a Newtonian straight line but a line that is seen in Newtonian space as a curved line. Moreover, the factor causing the curvature of the line is the existence of mass. According to this description, the gravitational attraction is only the result of a geometrical feature. For example, we think that the Sun attracts the Earth, and that is why the Earth does not continue in a straight line but revolves around the Sun; in effect what happens is that the Sun distorts the space in such a way that the elliptical orbit of the Earth is actually a “straight line” in the sense that it is the shortest route in the geometry of the space. Is this merely playing with words, or are we dealing with a physical feature that our senses did not perceive previously? The test will be whether the theory explains things that cannot be explained any other way, and it will be even more convincing if it predicts new effects.

One aspect that Einstein explained using the new geometry was the very small changes discovered in the path of the planet Mercury. Astronomers had proposed other explanations for those variations, such as the effect of another as yet undiscovered planet. Einstein also proposed a new prediction. If it is geometry that is the determining factor, then a physical body on which ordinary gravity ought not to act would follow the curved line that Einstein forecasts and would not follow a Newtonian straight line. Light itself is such a physical object. If physical space curves around the Sun, then the light of a star that reaches us and that passes around the Sun will reach us on a curved route and will seem to us to be in a different location. We do not generally see the light of stars coming from the direction of the Sun. The best time to identify and measure the direction from which the light is arriving is during a full eclipse of the Sun. A number of scientific expeditions tried to check Einstein's prediction over several years but failed either because of bad atmospheric conditions when the sky was hidden by clouds precisely at the time of the eclipse or because of political events, such as when an eclipse of the Sun occurred in the middle of a war between Germany and Russia and the expedition's astronomical equipment was confiscated by the Russian authorities who were concerned about espionage.

The confirmation came on May 29, 1919, when there was one of the longest total eclipses of the Sun, lasting almost seven minutes. The eclipse started in Brazil and moved to South Africa. Two expeditions were organized by the British Royal Society; one went to Brazil, and the other to a small island off the coast of South Africa. Both of them managed to measure and corroborate Einstein's general theory of relativity. Later, doubts were expressed whether the equipment they had used was accurate enough to enable the conclusions they had reached to be drawn. In any event, since then the general theory of relativity has been confirmed many times. Einstein's equations correctly describe the geometry of the world. As was the case with the special theory of relativity, the general theory was also for many years relevant only to scientists. Today, with the wide use of outer space, for instance for GPSs (global positioning systems), effects that are part of general relativity are also relevant to engineering.

But even Einstein, the master of intuitive interpretations of mathematics, was not immune from the trickery of intuition. The common perception was that the gravitational force will, eventually, cause the universe the collapse. Einstein's intuition told him that the universe is static, and he corrected the equation by adding a constant, the cosmological constant, that stabilizes the equations. Later, Edwin Hubble discovered that the universe is indeed expanding, and with a constant rate. Einstein removed the cosmological constant from the equations, referring to the addition of the constant as the biggest mistake in his scientific life. This can be interpreted as saying that, had he believed in the original equations, he could have predicted the expansion of the universe before it was discovered by analysis of experimental data. Recently it was discovered that the expansion itself is accelerated. This can be explained by putting back the cosmological constant into the equations, this time to account for the acceleration of the expansion. Thus, it could turn out that the removal of the constant from the equation was another mistake by Einstein.

Albert Einstein was by all accounts the most famous scientist of the modern era and, hence, the most written about. Here we will present just a few of the central facts about his life and work relevant to our account. Einstein was born in 1879 in the town of Ulm, then in the Kingdom of Württemberg, part of the German Empire, and he moved with his parents to Munich when he was one year old. As a child and youth, he did not stand out as a student, but nor did he lag behind (as rumored later on). When he was fifteen years old, his family moved to Italy for economic reasons. Albert joined them but did not acclimatize well to the new environment, and he was sent to complete his secondary schooling in Aarau, in northern Switzerland. In 1896 he was accepted into the Swiss Federal Polytechnic (known today as the Swiss Federal Institute of Technology) in Zurich, and he graduated in 1900. In his university studies he did not shine either, mainly because he concentrated on subjects that interested him, physics, mathematics, and philosophy, and in these too he did not persevere with the studies themselves but invested his time and energy in independent reading. On completing his studies he tried for some years to obtain a teaching post, unsuccessfully, and eventually, in 1903, was given the position of examiner in the Swiss patent office in Bern. There he had to evaluate many patent applications for electromagnetic devices, whose uses in engineering were constantly increasing.

Einstein had come across Maxwell's theory in his studies, and he continued to be interested in it and to involve himself in the scientific side of the theory, as he did in other scientific and philosophical subjects, but not in any formal academic framework. At the same time, he studied and carried out research at the University of Zurich, being awarded his doctorate in 1905. In that same year, while still working as a patents examiner, he published four groundbreaking papers that have left a deep imprint on science. The first paper gave an explanation of the photoelectric effect, and we will return to this in the next section. Two of the papers dealt with what is now referred to as the special theory of relativity: the first dealt with the laws of mechanics that were derived from the new geometry that we described above, and the second with the equivalence of energy and matter. The fourth paper in the series, which resulted in the year 1905 becoming known as Einstein's annus mirabilis (miracle year), described the mathematical basis of the motion of particles similar to Brownian motion. This is the random motion of microscopic particles reported in various situation, named after the eighteenth-century Scottish botanist Robert Brown. This paper of Einstein's served as the springboard for the mathematical subject called random motion, a ramified area of mathematics still of active interest today.

These outstanding contributions brought Einstein broad academic acclaim, which led to his being offered an associate professorship at the University of Zurich. His reputation in the academic world did not filter through to the general public fast enough. When Einstein resigned from the patent office in 1909, four years after his annus mirabilis, explaining that it was because of the offer of a teaching post in the University of Zurich, his superior in the office reacted by saying, “Einstein, stop fooling around. Tell me the real reason for your resignation.” At that time he had already started to work on the theory of gravity, to which he devoted about ten years that ended with the publication in 1916 of his paper presenting the general theory of relativity. Meanwhile he served for short periods as a professor at the University of Prague and at the Swiss Polytechnic in Zurich.

In 1913 he received a personal invitation from two of the best-known scientists in the world at that time, the physicist Max Planck and the chemist Walther Nernst, who came to Zurich to persuade Einstein to accept the position of head of the Kaiser Wilhelm Institute of Physics in Berlin, and Einstein moved there in 1914. As stated in the previous section, in 1919 the general theory of relativity was confirmed, an event that spread Einstein's fame worldwide.

He was awarded the 1921 Nobel Prize in Physics for his contribution to the understanding of the photoelectric effect, not for the theory of relativity. The Royal Swedish Academy of Sciences of course does not publish reasons why it does not award the prize for a particular achievement, but unofficially it was explained that according to the will of Alfred Nobel, the prize is supposed to be awarded for achievements of practical value to the welfare of humanity, and the theory of relativity was not considered to have such value. Obviously even such a narrow guideline was incorrect with regard to the theory of relativity. According to other rumors, some of the members of the prize committee of the Royal Swedish Academy of Sciences, although recognizing the greatness of the achievement, were still not convinced that the theory of relativity was correct.

Einstein spent his time in Berlin mainly in research in attempts to find a single theory that would explain both the mechanics of gravity and the quantum theory that had been developed in the meantime. He continued with these attempts, without much success, up to his final years in the United States, where he had moved due to the Nazis’ rise to power in 1933. Luckily he was not in Germany (he was on a visit to the United States) at the time of the regime change; under the Nazi government his property was confiscated, he lost his German citizenship, and his theory was declared an incorrect Jewish theory. He spent some time at the California Institute of Technology (also known as Caltech) in Pasadena, not far from Los Angeles, and later joined the Institute for Advanced Study in Princeton, New Jersey. He became an American citizen in 1940.

Experimental confirmation of mass-energy equivalence came only in the 1930s. The outbreak of World War II led to the accelerated development of the technique of converting mass to energy, a development whose peak was the manufacture of the atom bomb. Dropping the atom bombs on Hiroshima and Nagasaki led to the end of the war.

In general Einstein distanced himself from politics, but he did not hesitate to express his liberal pacifist views. Nevertheless, in the Second World War he signed a letter in favor of the development of a nuclear bomb in order to achieve nuclear power before the Germans. Einstein, who was a secular Jew all his life, identified with his Jewishness and the Jewish people, supported the creation of the State of Israel, and was even invited to become its president after the death of its first president, Chaim Weizmann. He turned the offer down politely, as he did not consider himself suitable to the position, stating that he lacked “both the natural aptitude and the experience to deal properly with people and to exercise official functions.” He died in Princeton in 1955.